Space is big. It is very, very big. On the currently most favored
cosmological theories, we are living in an infinite world, a world that
contains an infinite number of planets, stars, galaxies, and black holes.
This is an implication of most “multiverse theories”, according to which
our universe is just one in a vast ensemble of physically real universes.
But it is also a consequence of the standard Big Bang cosmology, if
combined with the assumption that our universe is open or flat, as recent
evidence suggests it is. An open or flat universe – assuming the simplest
topology[1] – is spatially infinite at any time
and contains infinitely many planets etc.[2]

Philosophical investigations relating to the vastness of the cosmos
have focused on the fine-tuning of our universe. “Fine-tuning” refers
to the alleged fact that the laws of physics are such that if any of
several physical constants had been even slightly different, then life
would not have existed. A philosophical cottage industry has arisen
from the controversies surrounding issues such as whether fine-tuning
is in some sense “improbable”, whether it should be regarded as surprising[3],
whether it calls out for explanation (and if so whether a multiverse
theory could explain it[4], whether it suggests ways in which current physics
is incomplete[5],
or whether it is evidence for the hypothesis that our universe was designed[6].

Here I wish instead to address a more fundamental problem: How can
vast-world cosmologies have any observational consequences
at all? I will show that these cosmologies imply, or give a very
high probability to, the proposition that every possible observation
is in fact made. This creates a challenge: if a theory is such that
for any possible human observation that we specify, the theory says
that that observation will be made, then how do we test the theory?
What could possibly count as negative evidence? And if all theories
that share this feature are equally good at predicting the data we will
get, then how can empirical evidence distinguish between them?

I call this a “challenge” because current cosmological theories clearly
do have connections to observation. Cosmologists are constantly modifying
and refining theories in light of empirical findings, and they are presumably
not irrational in doing so. But it is a philosophical problem to account
for how this is possible.

One lesson that will emerge is that we must be careful about how we
construe the evidence. We know not only that such-and-such observations
are made (which we shall show is impotent as a basis for evaluating
Big World theories): we also know that such-and-such observations are
made by us. This indexical de se component of our evidence
turns out to be crucial to cosmology, and recognizing this is the first
step to the solution that I shall propose.

The second step is to formulate a new methodological principle that
describes the probabilistic evidential bearing of (partly) indexical
information on non-indexical hypotheses.

With the expanded evidence base and the new rule, we can explain how
Big World theories are testable. We will also hint at how the epistemological
theory we outline is useful in other areas of philosophy and scientific
methodology.

But first, let us study in more detail how things go wrong if we construe
the evidence non-indexically, in the form “Such-and-such an observation
is made”. We can be generous and take “an observation” in a broad sense
to include the total phenomenological content present in the observer’s
mind. We do not, however, at this stage take “observing” as success
verb, implying the veracity of observations; but rather, we assume an
internal reading of the evidence. This assumption will later be relaxed.

I. THE CONUNDRUM

Consider a random phenomenon, for instance Hawking radiation. When
black holes evaporate, they do so in a random manner such that for any
given physical object there is a finite (although extremely small) probability
that it will be emitted by any given black hole in a given time interval.
Such things as boots, computers, or ecosystems have some finite probability
of popping out from a black hole. The same holds true, of course, for
human bodies and human brains in particular states.[7]
Assuming that mental states supervene on brain states, there is thus
a finite probability that a black hole will produce a brain in a state
of making any given observation. Some of the observations made by such
a brains will be illusory and some will be truthful. For example, some
brains produced by black holes will have the illusory of experience
of reading a measurement device that does not exist. Other brains, with
the same experiences, will be making veridical observations – a measurement
device may materialize together with the brain and may have caused the
brain to make the observation. But the point that matters here is that
any observation we could make has a finite probability of being produced
by any given black hole.

The probability of anything macroscopic and organized appearing
from a black hole is of course minuscule. The probability of a given
conscious brain-state being created is tinier still. Yet even a low-probability
outcome has a high probability of occurring if the random process is
repeated often enough. And that is precisely what happens in our world,
if the cosmos is very vast. In the limiting case where the cosmos contains
an infinite number of black holes, the probability of any given observation
being made is one.[8]

There are good grounds for believing that our universe is open and
contains an infinite number of black holes. Therefore, we have reason
to think that any possible human observation is in fact instantiated
in the actual world.[9] Evidence for the
existence of a multiverse would only add further support to this proposition.

It is not necessary to invoke black holes to make this point. Any random
physical phenomenon would do. It seems we don’t even have to limit the
argument to quantum fluctuations. Classical thermal fluctuations could,
presumably, in principle lead to the molecules in a gas cloud containing
the right elements to spontaneously bump into each other so as to form
a biological structure such as a human brain.

The problem is that it seems impossible to get any empirical evidence
that could distinguish between various Big World theories. For any observation
we make, all such theories assign a probability of one to the
hypothesis that that observation is made. That means that the fact that
the observation is made is no reason whatever for preferring one of
these theories to the others. Experimental results appear totally irrelevant.[10]

We can see this formally as follows. Let B be the proposition
that we are in a Big World, defined as one that is big enough and random
enough to make it highly probable that every possible human observation
is made. Let T be some theory that is compatible with B,
and let E be some proposition asserting that some specific observation
is made. Let P be an epistemic probability function. Bayes’s
theorem states that

P(T|E&B) = P(E|T&B)P(T|B)
/ P(E|B).

In order to determine whether E makes a difference to the probability
of T (relative to the background assumption B), we need
to compute the difference P(T|E&B) -
P(T|B). By some simple algebra it is easy to see
that

P(T|E&B) - P(T|B)0 if and only if
P(E|T&B)P(E|B).

This means that E will fail to give empirical support to T
(modulo B) if E is about equally probable given T&B
as it is given B. We saw above that P(E|T&B)P(E|B)1. Consequently,
whether E is true or false is irrelevant for whether we should
believe in T, given we know B.

To illustrate, let T2 be some perverse permutation
of an astrophysical theory T1 that we actually embrace.
T2 differs from the T1 by assigning
a different value to some physical constant. To be specific, let us
suppose that T1 says that the temperature of the cosmic
microwave background radiation is about 2.7 Kelvin (which is the observed
value) whereas T2 says it is, say, 3.1 K. Suppose
furthermore that both T1 and T2
imply that we are living in a Big World. One would have thought that
our experimental evidence favors T1 over T2.
Yet the above argument seems to show that this view is mistaken. Our
observational evidence supports T2 just as much as
T1. We really have no reason to think that the background
radiation is 2.7 K rather than 3.1 K.

II. IT’S NOT THE OLD POINT ABOUT
UNDERDETERMINATION OF THEORY BY DATA

At first blush, it could seem as if this simply rehashes the lesson,
made familiar by Duhem and Quine, that it is always possible to rescue
a theory from falsification by modifying some auxiliary assumption,
so that strictly speaking no scientific theory ever implies any observational
consequences. The above argument would then merely have provided an
illustration of how this general result applies to cosmological theories.
However, this would be to miss the point.

If the argument given above is correct, it establishes a much more
radical conclusion. It purports to show that all Big World theories
are not only logically compatible with any observational evidence, but
they are also perfectly probabilistically compatible. They all
give the same conditional probability (namely one) to every observation
statement E defined as above. This entails that no such observation
statement can have any bearing, whether logical or probabilistic,
on whether the theory is true. If that were the case, it would not seem
worthwhile to make astronomical observations if what we are interested
in is determining which Big World theory to favor. The only reasons
we could have for choosing between such theories would be either a priori
(simplicity, elegance, etc.) or pragmatic (such as ease of calculation).

Nor is the argument making the ancient statement that human epistemic
faculties are fallible, that we can never be certain that we are not
dreaming or are brains in a vat. No, the point here is not that such
illusions could occur, but rather that we have reason to believe
that they do occur, not just some of them but all possible ones.
In other words, we can be fairly confident that the observations we
make, along with all possible observations we could make in the future,
are being made by brains in vats and by humans that have spontaneously
materialized from black holes or from thermal fluctuations. The argument
would entail that this abundance of observations makes it impossible
to derive distinguishing observational consequences from contemporary
cosmological theories.

III. THE CONCLUSION IS A REDUCTIO

I trust that most readers will find this conclusion unacceptable. Cosmologists
certainly appear to be doing experimental work and modify their theories
in light of new empirical findings. The COBE satellite, the Hubble Space
Telescope, and other devices are showering us with data that have been
causing something of a renaissance in the world of astrophysics in recent
years. Yet the argument described above would show that the empirical
import of this information could never go beyond the humble role of
providing support for the hypothesis that we are living in a Big World,
for instance by showing that the universe is open. Nothing apart from
this one fact could be learnt from such observations. Once we have established
that the universe is open and infinite, then any further work in observational
astronomy would be a waste of time and money.

Worse still, the leaky connection between theory and observation in
cosmology spills over into other domains. Since nothing hinges on how
we defined T in the derivation above, the argument can easily
be extended to prove that observation does not have a bearing on any
scientific question so long as we assume that we are living in a Big
World.[11]

This consequence is absurd, so we should look for a way to mend the
methodological pipeline and restore the flow of testable observational
consequences from Big World theories. How can we do that?

IV. GIVING UP THE INTERNAL CONSTRUAL
OF “OBSERVATION” DOESN’T SAVE US

Suppose we give up the internal construal of “observation” and instead
take the term as a success verb, so that observing, say, a blue table
implies that there is a blue table that is causally responsible for
the observation. Suppose further that we couple this with the postulation
that we are entitled (and perhaps even required) to have a prior credence
function that strongly favors the hypothesis that we for the most part
really do observe (in the success sense) what it seems to us that we
are observing. Then it might appear as if we have an exit from our predicament.
(Alternatively, we could formulate this escape plan by sticking to the
original internal definition of “observation” and adding the postulate
that our prior credence functions should strongly favor the veridicality
of our observations.)

However, even setting aside foundationalist scruples, the proposed
solution doesn’t get us out of the pickle.

To see this, consider that observers are not the only things that have
a finite probability of being generated in random systems. On the same
ground that we should expect human observers in all possible states
to be ejected from black holes or to form from vastly improbable thermal
fluctuations, we should also expect all physically possible local environments
to spring forth. So not only are there observers having all sorts of
illusions (of seeing a blue table or reading a measurement apparatus)
but additionally there are observers making all sorts of veridical
observations (actually seeing a blue table or reading off instruments
in each of their possible output states). Consequently, even if we assume
our observations to be veridical, we are still left with the problem
that our current best theories give probability one to the existence
of all possible such observations together with their truth-making
local environments. (See Figure 1). We can even press on to the
conclusion that for any possible human observation, there may be habitats
in which that observation is appropriately caused by the observed
object and in which the observer’s perceptions in general track her
surroundings.[12]

A qualification is due. While small-scale environments, e.g. ones that
include tables and measuring apparatuses, are on a par with human bodies,
it is not clear that very large systems such as galactic superclusters
could be produced by any of the random processes that we have discussed.
If we stipulate that we are making veridical observations of these mega-scale
entities, we could thus salvage the testability of some aspects of cosmological
theories that concern these large-scale entities. Yet this would be
of little avail since it would not rescue the rest of our epistemic
practices, which deal with medium-sized and small things. Observations
of such items would still be subject to the charge of being radically
irrelevant to our theories about the world, modulo the Big World hypothesis.[13]

A further shortcoming of the proposal (apart from the fact that it
doesn’t work) is that it doesn’t tell us anything about the defeasibility
conditions of the purported principle that you should be strongly biased
in favor of the veridicality of your observations. Clearly, there are
cases where it would be unreasonable to believe that one’s observations
are veridical. For example, if you knew that almost all observers in
your current situation (tucked in, let’s say, between the bedsheets
in a detox unit with the sensation of bugs crawling under your skin)
were hallucinating, then you should not believe that your current
observations are veridical, unless you had additional information defeating
that conclusion. A satisfactory account of the Big World case
ought to have at least something to say about why the presence of lots
of hallucinating and otherwise misled observers in Big Worlds does not
undermine our confidence in the reliability of our own observations
while the contrary holds specifically for clients in the methadone clinic
and other such special situations.

So if an externalist construal of the evidence is not the answer,
what is?

V. RESTORING THE FLOW OF TESTABLE
CONSEQUENCES VIA A LIMITED INDIFFERENCE PRINCIPLE OVER DE SE
STATEMENTS

It may seem as if our troubles originate from the somewhat “technical”
point that in a large enough cosmos, every observation will be made
by some freakish observers here and there. It remains the case,
however, that those observers are exceedingly rare and far between.
For every observation made by a freak observer spontaneously materializing
from Hawking radiation or thermal fluctuations, there are trillions
upon trillions of observations made by regular observers who have evolved
on planets like our own and who make veridical observations of the universe.
Why can we not solve the problem, then, by saying that although all
these freak observers exist and are suffering from various illusions
(or are making veridical but unrepresentative observations), it is highly
unlikely that we are among their numbers? Then we should think,
rather, that we are very probably one of the regular observers whose
observations reflect reality. We could safely ignore the freak observers
and their illusions and misleading perceptions in most contexts when
doing science.

In my view, this response suggests the right way to proceed. Because
the freak observers are in such a tiny minority, their observations
can be disregarded for most purposes. It is possible that we
are freak observers – we should assign to that hypothesis some finite
probability, but such a tiny one that it does not make any practical
difference.

If we want to run with this idea, it is crucial that we construe our
evidence differently than we did above. If our evidence is simply “Such
and such an observation is made” then the evidence has probability one
given any Big World theory – and we ram our heads straight into the
problems I described. But if we construe our evidence in the more specific
form “We are making such and such observations” then we have
a way out. For we can then say that although Big World theories make
it probable that some such observations be made, they need not make
it probable that we should be the ones making them.

Let us therefore define:

E’ := “Such and such observations are made by us”

E’ contains an indexical de se component that the original evidence-statement
we considered, E, did not. E’ is logically stronger than
E. The rationality requirement that one should take all relevant
evidence into account dictates that in case E’ leads to different
conclusions than does E, then it is E’ that determines
what we ought to believe.

A question that now arises is how to determine the evidential bearing
that statements of the form of E’ have on cosmological theories.
Using Bayes’s theorem, we can turn the question around and ask, how
do we evaluate P(E’|T&B), the conditional
probability that a Big World theory gives to us making certain observations?
The argument in the foregoing sections showed that if we hope to be
able to derive any empirical implications from Big World theories, then
P(E’|T&B) should not generally be set
to unity or close to unity. P(E’|T&B)
must take on values that depend on the particular theory and the particular
evidence that we are we are considering. Some theories T are
supported by some evidence E’; for these choices P(E’|T&B)
is relatively large. For other choices of E’ and T, the
conditional probability will be much smaller.

To be concrete, consider the two rival theories T1
and T2 about the temperature of the cosmic microwave
background. Let E’ be the proposition that we have made those
observations that cosmologists innocently take to support T1.
E’ includes readings from radio telescopes etc. Intuitively,
we want P(E’|T1&B) > P(E’|T2&B).
That inequality must be the reason why cosmologists believe that the
background radiation is in accordance with T1 rather
than T2, since a priori there is no ground for assigning
T1 a substantially greater probability than T2.

A natural way to achieve this result is by postulating that we should
think of ourselves as being in some sense “random” observers. Here we
use the idea that the essential difference between T1
and T2 is that the fraction of observers that
would be making observations in agreement with E’ is enormously
greater on T1 than on T2. If we
reason as if we were randomly selected samples from the set of all observers,
or from some suitable subset thereof, then we can explicate the conditional
probability P(E’|T&B) in terms of the
expected fraction of all observers in the reference class that the conjunction
of T and B says would be making the kind of observations
that E’ says that we are making. As we shall see, this postulate
enables us to conclude that P(E’|T1&B)
> P(E’|T2&B).

Let us call this postulate the Self-Sampling Assumption:

(SSA) Observers should reason as if they were a random sample from
the set of all observers in their reference class.

The general problem of how to define the reference class is complicated,
and I shall not address it here. For the purposes of this paper we can
think of the reference class as consisting of all observers who will
ever have existed. We can also assume a uniform sampling density over
this reference class. Moreover, it simplifies things if we set aside
complications arising from assigning probabilities over infinite domains
by assuming that B entails that the number of observers is finite,
albeit such a large finite number that the problems described above
obtain. These assumptions help us focus on basic principles.

Here is how SSA supplies the missing link needed to connect theories
like T1 and T2 to observation. On
T2, the only observers who observe an apparent temperature
of the cosmic microwave background CMB2.7 K are those who
either have various sorts of rare illusions (for example because their
brains have been generated by black holes and are therefore not attuned
to the world they are living in) or happen to be located in extremely
atypical places (where e. g. a thermal fluctuation has led to a locally
elevated CMB temperature). On T1, by contrast, almost
every observer who makes the appropriate astronomical measurements and
is not deluded will observe CMB2.7 K. A much greater
fraction of the observers in the reference class observe CMB2.7 K if T1
is true than if T2 is true. By SSA, we consider ourselves
as random observers; so it follows that on T1 we would
be much more likely to find ourselves as one of those observers who
observe CMB2.7 K than we would
on T2. Therefore, P(E’|T1&B)
>> P(E’| T2&B). Supposing
that the prior probabilities of T1 and T2
are roughly the same, P(T1)P(T2),
it is then trivial to derive via Bayes’s theorem that P(T1|E’&B)
> P(T2|E’&B). This vindicates
the intuitive view that we do have empirical evidence that favors T1
over T2.

The job that SSA is doing in this derivation is to enable the step
from a proposition about fractions of observers to propositions about
corresponding probabilities. We get the propositions about fractions
of observers by analyzing T1 and T2
and combining them with relevant background information B; from
this we conclude that there would be an extremely small fraction of
observers observing CMB2.7 K given T2
and a much larger fraction given T1. We then consider
the evidence E’, which is that we are observing CMB2.7 K. SSA authorizes
us to think of the “we” as a kind of random variable ranging over the
class of actual observers. From this it then follows that E’
is more probable given T1 than given T2.
But without assuming SSA, all we can say is that a greater fraction
of observers observe CMB2.7 K if T1
is true, and at that point the argument would grind to a halt. We could
not reach the conclusion that T1 is supported over
T2. For this reason I propose that SSA, or something
like it, be adopted as a methodological principle.

It may seem mysterious how probabilities of this sort can exist – how
can we possibly make sense of the idea that there was some chance that
we might have been other observers than we are? However, what I am suggesting
here is not the existence of some objective, or physical, chances. I
am not suggesting that there is a physical randomization mechanism,
a cosmic fortune wheel as it were, that assigns souls to bodies in a
stochastic manner. Rather, we should think of these probabilities as
epistemic. They are part of a proposal explicating the epistemic
relations that hold between theories (such as T1 and
T2) and evidence (such as E’) containing a
de se component.We can view
SSA as a kind of restricted indifference principle that applies to credences
over de se propositions, or sets of centered possible worlds in the
Quinean terminology. The status of SSA could also be regarded as in
some respects akin to that of the David Lewis’s Principal Principle[14], which expresses
a connection between physical chance and epistemic credence. Crudely
put, the Principal Principle says that if you know that the objective
(physical) chance of some outcome A is x%, then you should
assign a credence of x% to A (unless you have additional
“inadmissible” information). Analogously, SSA can be read as saying
that if you know that a fraction x% of all observers in your
reference class are in some type of position A, then you should
assign a prior credence of x% to being in a type-A position.
This prior credence must, of course, be conditionalized on any other
relevant information you have in order to get the posterior credence,
i.e. the degrees of belief you should actually have given all you know.
Thus, after conditionalizing on the observation that CMB2.7 K, you get, trivially,
a posterior function that assigns zero credence to the hypothesis that
you are an observer that observes CMB3.1 K. But it is
the higher conditional prior credence (according to SSA) of observing
that CMB2.7 K given T1
than given T2 that renders it the case that conditionalizing
on this observation preferentially supports T1.

VI. AN ILLUSTRATION

We can illustrate how SSA works by a simple thought experiment.

Blackbeards and Whitebeards.

In an otherwise empty world there are three rooms. God tosses a fair
coin and creates three observers as a result, placing them in different
rooms. If the coin falls heads, He creates two observers with black
beards and one with a white beard. If it falls tails, it is the other
way around: He creates two whitebeards and one blackbeard. All observers
are aware of these conditions. There is a mirror in each room, so
observers know the color of their own beard. You find yourself in
one of the rooms and you see that you have a black beard. What credence
should you give to the hypothesis that the coin fell heads?

The situation is depicted in Figure 2.

Because of the direct analogy to the cosmology case, we know that the
answer must be that you should assign a greater credence to Heads
than to Tails. Let us apply SSA and see how we get this result.

From the setup, we know that the prior probability of Heads
is 50%. This is the probability you should assign to Heads before
you have looked in the mirror and thus before you know your beard color.
That this probability is 50% follows from the Principal Principle together
with the fact you know that the coin toss was fair. We thus have

Next we consider the conditional probability of you observing that
you have black beard given a specific outcome of the toss. If the coin
fell heads, then two out of three observers observe themselves having
a black beard. If the coin fell tails, then one out of three observe
having a black beard. By SSA, you reason as if you were a randomly sampled
observer, giving

.

Using Bayes’s theorem, we can then calculate the conditional probability
of Heads given that you have a black beard:

.

After looking in the mirror and learning that your beard is black you
should therefore assign a credence of to Heads
and to Tails.

This result mirrors that of the cosmology example. Because one theory
(T1, Heads) entails that a greater fraction
of all observers are observing what you are observing (E’, Black)
than does another theory (T2, Tails), the former
theory obtains preferential support from your observation.

VII. SUMMARY: WE NEED A METHODOLOGY
FOR EVIDENCE WITH A DE SE COMPONENT

Big World theories, popular in contemporary cosmology, engender a peculiar
methodological problem: because they say the world is very big and somewhat
stochastic, they imply (or make it highly probable) that every possible
human observation is made. The difficulty is that it is unclear how
we could ever have empirical reasons for preferring one such theory
to another, since they all seem to fit equally well with whatever we
observe. This skeptical threat is different from and much more radical
than the problem of underdetermination of theory by data associated
with Duhem and Quine. And if left unfixed, the broken connection between
observation and theory spills over from cosmology into other domains.

We saw that the leakage cannot be stopped even by blocking all consideration
given to the possibility of illusory observations, because the maverick
observations made in Big Worlds include veridical ones as well as illusions.
Instead, we proposed to repair methodology by means of a new epistemic
principle, the Self-Sampling Assumption, which takes into account the
de se component of our evidence. This principle connects Big World theories
to observation in an intuitively plausible way and vindicates the practices
of cosmologists who test hypotheses against experimental findings.

The Self-Sampling Assumption has implications in other problem areas
in science and philosophy. It can be seen as an explication of the anthropic
principle, understood in the original spirit of by Brandon Carter, a
theoretical physicist whose seminal work opened the door to a systematic
exploration of observation selection effects.[15]
Observation selection effects are a kind of bias that may be present
in our data that is not due to limitations in our measurement apparatuses
but to the fact that our data are preconditioned on the existence of
a suitably positioned observer to “have” the data (and to build the
instruments in the first place). Carter investigated the relevance of
observation selection effects for attempts to evaluate the bearing of
our current evidence on questions such as how improbable it is for complex
life forms to evolve on a given Earth-like planet or how many critical
improbable steps were involved in our evolution.[16]
To illustrate, take one of the simplest points Carter made: Even if
a theory says that the probability for an Earth-like planet of giving
rise to intelligent life is small, the theory will still perfectly fit
our observation of intelligent life having evolved on this planet provided
that the total number of Earth-like planets is large enough for it to
have been probable, according to the theory, that intelligent life should
arise somewhere.

Similar modes of reasoning are invoked in some discussions of no-collapse
versions of quantum mechanics[17] and, as hinted at in the introduction,
they play a central role in the debate about the significance of the
apparent fine-tuning of our universe and the capacity of multiverse
theories to explain it. Even an application to traffic planning has
been discovered.[18] On the more theoretical side,
we have game theoretic problems involving imperfect recall, such as
the Absent-Minded Driver problem[19]
and its philosophical, more purely epistemic analogue, the Sleeping
Beauty problem[20].

What these various topics have in common is that they involve the assignment
of conditional credences to statements of the form “I make such and
such observations given that the world is such and such.”[21]
In other words, they involve the evaluation of a de se component of
our evidence: our knowledge that we are the ones making a certain
observation or that we are the ones who have a certain piece
of (otherwise non-indexical) evidence. Our duty to objectivity must
not be misunderstood as a license to ignore de se clues. The considerations
advanced in this paper impose constraints on what can count as a satisfactory
methodology for fashioning knowledge out of this indexical part our
epistemic raw material. Such a methodology, a general theory of observation
selection effects and its various scientific and philosophical applications,
is something I have attempted to set forth elsewhere.[22]

[1] I.e. that space is
simply connected. There is a recent burst of interest in the possibility
that our universe might be multiply connected, in which case it could
be both finite and hyperbolic. A multiply connected space could lead
to a telltale pattern consisting of a superposition of multiple images
of the night sky seen at varying distances from Earth (roughly, one
image for each lap around the universe that the light has traveled).
Such a pattern has not been found, although the search continues. For
an introduction to multiply connected topologies in cosmology, see M.
Lachièze-Rey and J.-P. Luminet, J.-P., “Cosmic Topology,” Physics
Reports, 254(3) (1995): 135-214.

[2] A widespread misconception
is that the open universe in the standard Big Bang model becomes spatially
infinite only in the temporal limit. The observable universe
is finite, but only a small part of the whole is observable (by us).
One fallacious intuition that might be responsible for this misconception
is that the universe came into existence at some spatial point in the
Big Bang. A better way of picturing things is to imagine space as an
infinite rubber sheet, and gravitationally bound groupings (such as
stars and galaxies) as buttons glued on to it. As we move forward in
time, the sheet is stretched in all directions so that the separation
between the buttons increases. Going backwards in time, we imagine the
buttons coming closer together until, at “time zero”, the density of
the (still spatially infinite) universe becomes infinite everywhere.
See e.g. J. L. Martin, General Relativity (London: Prentice Hall,
1995).

[7] See e.g. S. Hawking
and W. Israel, eds., General Relativity: An Einstein Centenary Survey
(Cambridge University Press, 1979): “[I]t is possible for a black hole
to emit a television set or Charles Darwin” (p. 19). To avoid making
a controversial claim about personal identity, Hawking and Israel ought
to have weakened this to “… an exact replica of Charles Darwin”. But
see also G. J. Belot et al., “The Hawking Information Loss Paradox:
The Anatomy of a Controversy,” British Journal for the Philosophy
of Science 50(2) (1999): 189-229.

[8] In fact, there is
a probability of unity that infinitely many tokens of each observation-type
will appear. But one of each suffices for present purposes.

[9] I restrict the assertion
to human observations in order to avoid questions as to whether
there may be other kinds of possible observations that perhaps could
have infinite complexity or be of some alien or divine nature that does
not supervene on stuff that is emitted from black holes – such stuff
is physical and of finite size and energy.

[11] And were it really
true that we have no means of testing Big World theories, then it is
not even clear that the empirical support we currently have for such
theories could be maintained. Such theories would seem self-undermining
in that they would say of their own evidence, in effect, that it was
not to be trusted.

[12] I want to emphasize
that the problem is not that there is some massive inconsistency of
contradictory observations. To assert the existence of all possible
human observations is not inconsistent, since the observations may be
illusory. Moreover, even if all the observations were asserted to be
veridical, it would still be no inconsistency, since the various diverse
properties that are being observed may be instantiated at different
places, just as a tie can be both blue and yellow (although not at the
same spot at the same time). Rather, the problem is how to derive testable
predictions given our inability to observationally locate ourselves
in a Big World, which is rather analogous to seeing a yellow spot through
a microscope and not knowing which part of the hypothesized tie we are
looking at.

[13] We may also note
that there are some (speculative) theories according to which even the
largest structures that we see are not large enough to escape the problem
(e.g. M. Tegmark, “Does the universe in fact contain almost no information?”
Foundations of Physics Letters, 9(1) (1996): 25-42). Moreover,
there are many much less extreme theories, such as chaotic inflation
theory (see e.g. A. Linde, “Inflation with variable Omega,” Physics
Letters B,351 (1995): 99-104), according to which observers
are observing a wide range of different values of some physical constant
and parameters, not because the observers have illusions or live in
habitats that originate from black holes or the like, but because the
“constants” and parameters vary over vast cosmic distances or epochs.

[21] Or in some cases,
the analogous temporal construction: “I make such and such observations
now given that the world is such and such.”

[22]Anthropic Bias:
Observation Selection Effects in Science and Philosophy (New York:
Routledge, 2002). A theory of observation selection effects must walk
a fine line in order to cater to legitimate scientific needs while avoiding
philosophical paradoxes, of which a great number lie in ambush. Incidentally,
the Self-Sampling Assumption is, in my view, a mere derivative of a
more powerful principle, and it is only valid in special cases.